# laws of logarithms

• October 28th 2006, 08:19 AM
Yogi_Bear_79
laws of logarithms
Please help with this example to apply the laws of logarithms to simplify the function $f(x) =ln \sqrt\frac{9-x^2}{4+x^2}$. Then find it’s derivative.
• October 28th 2006, 08:27 AM
Quick
Quote:

Originally Posted by Yogi_Bear_79
Please help with this example to apply the laws of logarithms to simplify the function $f(x) =ln \sqrt\frac{9-x^2}{4+x^2}$. Then find it’s derivative.

I can't help you with the derivative, but here's the simply part:

you have: $\ln\sqrt{\frac{9-x^2}{4+x^2}}$

rewrite: $\ln\left(\left(\frac{9-x^2}{4+x^2}\right)^{\frac{1}{2}}\right)$

Thus: $\frac{1}{2}\ln\frac{9-x^2}{4+x^2}$

rewrite: $\frac{1}{2}\ln\frac{(3+x)(3-x)}{4+x^2}$

It is possible to go farther but I don't know how that might help...
• October 28th 2006, 08:35 AM
Soroban
Hello, Yogi_Bear_79!

Do you know the Laws of Logarithms?

Quote:

Apply the laws of logarithms to simplify the function $f(x) = \ln\sqrt\frac{9-x^2}{4+x^2}$
Then find its derivative.

We have: . $f(x) \;= \;\ln\sqrt{\frac{9-x^2}{4+x^2}} \:=\:\ln\!\left(\frac{9-x^2}{4+x^2}\right)^{\frac{1}{2}} \;=\;\frac{1}{2}\cdot\ln\!\left(\frac{9-x^2}{4+x^2}\right)$

Then: . $f(x)\;=\;\frac{1}{2}\bigg[\ln(9-x^2) - \ln(4 + x^2)\bigg]$

Now differentiate it . . .

• October 28th 2006, 08:37 AM
Quick
Quote:

Originally Posted by Soroban

Then: . $f(x)\;=\;\frac{1}{2}\bigg[\ln(9-x^2) - \ln(4 + x^2)\bigg]$

Wait, so $\log\frac{a}{b}=\log a-\log b$? That's useful...
• October 28th 2006, 12:23 PM
topsquark
Quote:

Originally Posted by Quick
Wait, so $\log\frac{a}{b}=\log a-\log b$? That's useful...

Yes it's true and yes it's useful! :)

-Dan
• October 28th 2006, 01:04 PM
earboth
Quote:

Originally Posted by Quick
...That's useful...

Hi,

I've attached a pdf-file with the logarithm rules and the corresponding power rules. Maybe this is useful for you too.

EB